Reversing Borcherds Lifts: A Survey

This is a survey on our results towards the reversing of the multiplicative theta lift of Borcherds. First results on this topic had been obtained by Bruinier, proving that meromorphic automorphic forms F on the orthogonal group G = O(2, n + 2) with special divisor are Borcherds lifts. Holomorphic automorphic forms on G are Borcherds lifts if and only if they have a certain symmetry property. This leads to several applications. Special divisors (linear combinations of Heegner divisors) can be characterized by a symmetry property among all effective principal divisors. This gives a new proof and a generalization of parts of Bruinier’s result. We obtain recursion formulas for the Fourier-Jacobi coefficients of a Borcherds lift. Hence we have a direct link between Fourier-Jacobi coefficients and divisors.